Monochromatic Paths and at Most 2-Coloured Arc Sets in Edge-Coloured Tournaments

We call the tournament T an m-coloured tournament if the arcs of T are coloured with m-colours. If v is a vertex of an m-coloured tournament T, we denote by ξ(v) the set of colours assigned to the arcs with v as an endpoint. In this paper is proved that if T is an m-coloured tournament with |ξ(v)|≤2 for each vertex v of T, and T satisfies at least one of the two following properties (1) m≠3 or (2) m=3 and T contains no C₃ (the directed cycle of length 3 whose arcs are coloured with three distinct colours). Then there is a vertex v of T such that for every other vertex x of T, there is a monochromatic directed path from x to v. Received: April, 2003     

Preparation of uniform rich cholesterol unilamellar nanovesicles using CO2-expanded solvents

Nanoscopic and uniform unilamellar vesicles, rich in cholesterol, have been prepared by a new procedure, named "DELOS-SUSP", which is based on the depressurization of a cholesterol solution in CO2-expanded acetone into an aqueous solution containing a surfactant. The CO2 is used here as a cosolvent medium, allowing the straightforward preparation of vesicular systems with controlled size distribution, uniform shape, and stability unachievable by conventional mixing technologies. The resulting nanoscopic vesicular systems dispersed in water were characterized using dynamic light scattering, zeta potential, turbidity, and cryogenic transmission electron microscopy. The influence of operational parameters of this new methodology on the physicochemical characteristics of the vesicular systems is also reported.     

Mosaic CuI−CuII−InIII 2D Perovskites: Pressure-Dependence of the Intervalence Charge Transfer and a Mechanochemical Alloying Method

The perovskite (BA)₄[Cu^II(Cu^IIn^III)_0.5]Cl₈ ( 1_BA ; BA⁺=butylammonium) allows us to study the high-pressure structural, optical, and transport properties of a mixed-valence 2D perovskite. Compressing 1_BA reduces the onset energy of Cu^I/II intervalence charge transfer from 1.2 eV at ambient pressure to 0.2 eV at 21 GPa. The electronic conductivity of 1_BA increases by 4 orders of magnitude upon compression to 20 GPa, when the activation energy for conduction decreases to 0.16 eV. In contrast, Cu^II perovskites achieve similar conductivity at ≈50 GPa. The solution-state synthesis of these perovskites is complicated, with more undesirable side products likely from the precursor mixtures containing three different metal ions. To circumvent this problem, we demonstrate an efficient mechanochemical synthesis to expand this family of halide perovskites with complex composition by simply pulverizing together powders of 2D Cu^II single perovskites and Cu^IIn^III double perovskites.     

Richardson’s Theorem in <Emphasis Type="Italic">H</Emphasis>-coloured Digraphs

Let H be a digraph possibly with loops and D a finite digraph without loops whose arcs are coloured with the vertices of H (D is an H-coloured digraph). The sets V(D) and A(D) will denote the sets of vertices and arcs of D respectively. A directed path W in D is an H-path if and only if the consecutive colors encountered on W form a directed walk in H. A set \(N\subseteq \hbox {V}(D)\) is an H-kernel if for every pair of different vertices in N there is no H-path between them, and for every vertex \(u\in \hbox {V}(D){\setminus }N\) there exists an H-path in D from u to N. Let D be an m-coloured digraph. The color-class digraph of D, denoted by \({\mathscr {C}}_C(D\)), is the digraph such that: the vertices of the color-class digraph are the colors represented in the arcs of D, and \((i,j) \in A({\mathscr {C}}_C(D\))) if and only if there exist two arcs namely (u, v) and (v, w) in D such that (u, v) has color i and (v, w) has color j. Let \(W=(v_0, \ldots , v_n\)) be a directed walk in \({\mathscr {C}}_C(D)\), with D an H-coloured digraph, and \(e_i = (v_{i},v_{i+1})\) for each \(i \in \{0, \ldots ,n-1\}\). Let \(I = \{i_1, \ldots , i_k\}\) a subset of \(\{0, \ldots , n-1\}\) such that for 0 \(\le s \le n-1\), \(e_s \in \hbox { A}(H^c)\) if and only if \(s \in I\) (where \(H^c\) is the complement of H), then we will say that k is the \(H^c\)-length of W. Since V(\({\mathscr {C}}_C(D)) \subseteq \hbox {V}(H)\), the main question is: What structural properties of \({\mathscr {C}}_C(D)\), with respect to H, imply that D has an H-kernel? In this paper we will prove the following: If \({\mathscr {C}}_C(D)\) does not have directed cycles of odd \(H^c\)-length, then D has an H-kernel. Finally we will prove Richardson’s theorem as a direct consequence of the previous result.     

A Million Turnover Molecular Anode for Catalytic Water Oxidation

Molecular ruthenium‐based water oxidation catalyst precursors of general formula [Ru(tda)(Lⁱ)₂] (tda^2? is [2,2′:6′,2′′‐terpyridine]‐6,6′′‐dicarboxylato; L¹=4‐(pyren‐1‐yl)‐N‐(pyridin‐4‐ylmethyl)butanamide, 1 b ; L²=4‐(pyren‐1‐yl)pyridine), 1 c ), have been prepared and thoroughly characterized. Both complexes contain a pyrene group allowing ready and efficiently anchoring via π interactions on multi‐walled carbon nanotubes (MWCNT). These hybrid solid state materials are exceptionally stable molecular water‐oxidation anodes capable of carrying out more than a million turnover numbers (TNs) at pH 7 with an E_app=1.45 V vs. NHE without any sign of degradation. XAS spectroscopy analysis before, during, and after catalysis together with electrochemical techniques allow their unprecedented oxidative ruggedness to be monitored and verified.     

Fluorescence study of the fluidity and cooperativity of the phase transitions of zwitterionic and anionic liposomes confined in sol-gel glasses

The current work makes use of different fluorescent reporter molecules and fluorescent spectroscopic techniques to characterize the thermotropic, physical, and dynamical properties of large unilamellar liposomes formed from either 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) or 1,2-dimyristoyl-sn-glycero-3-[phospho-rac-glycerol] (DMPG) encapsulated in sol-gel matrixes. In particular, cooperativity of the phase transition is analyzed from steady-state fluorescence anisotropy of 1,6-diphenyl-1,3,5-hexatriene (DPH), the interfacial properties are studied by measuring the spectral shift of Laurdan, and the structural organization (heterogeneity) of the lipid bilayer is determined from the fluorescence lifetime of trans-parinaric acid (t-PnA). In addition, information regarding order and dynamical properties in the bulk hydrophobic core is obtained from time-resolved fluorescence anisotropy of t-PnA and 3-(4-(6-phenyl)-1,3,5-hexatrienyl)-phenylpropionic acid (PA-DPH). The spectroscopic study reveals that upon encapsulation, the basic thermodynamic properties as well as the fluidity of the lipid bilayer practically remain intact for DMPG liposomes but not for DMPC liposomes, whose lipid bilayer exhibits large gel-fluid heterogeneity. On the basis of these experimental results, electrostatic interactions between phospholipid polar heads and the porous surface of the host matrix seem to play a capital role for the preservation of the structural integrity of encapsulated bilayer.     

Strongly Emissive and Photostable Four‐Coordinate Organoboron N,C Chelates and Their Use in Fluorescence Microscopy

Six strongly fluorescent four‐coordinate organoboron N,C chelates containing an aryl isoquinoline skeleton were prepared. Remarkably, the fluorescence quantum yields reach values of up to 0.74 in oxygen‐free toluene. The strong B?N interaction was corroborated by the single‐crystal X‐ray analysis of two dyes. The intramolecular charge‐transfer character of the fluorophores was evidenced by solvatochromism studies and time‐dependent DFT calculations at the PCM(toluene)/CAM‐B3LYP/6‐311++G(2d,p)//PCM(toluene)/B3LYP/6‐311G(2d,p) level of theory. The compounds combine high chemical stability with high photostability, especially when equipped with electron‐donating substituents. The strong fluorescence and the large Stokes shifts predestine these compounds for use in confocal fluorescence microscopy. This was demonstrated for the imaging of the N13 mouse microglial cell line. Moreover, significant two‐photon absorption cross sections (up to 61 GM) allow the use of excitation wavelengths in the near‐infrared region (>800 nm).